Better results are obtained under the assumption of the Riemann Hypothesis. Harald Cramér proved that if the Riemann hypothesis holds, then the gap \(g_n\)  satisfies \(g_n=O(\sqrt{p_n}\log p_n)\) . Results of the form \(g_n=\mathcal {O}({p_n}^{\theta })\)  with different \(\theta <1\)  were given in the past. Among these where \(\theta \)  is close to \(1/2\)  are namely from [1]} \(\theta =7/12\) , Dr Brown gave an alternative proof, [2]} \(\theta =7/12\) , [3]} \(\theta =1051/1920\) , [4]} \(\theta =6/11\) , The best unconditional bound is known to Baker, Harman and Pintz, who proved the existence of \(x_0\)  such that there is a prime in the interval \([x, x+O(x^{21/40})]\)  for \(x>x_0\) , [5]}.
